In performing computer aided analysis and simulation of feedback circuits, it is often desirable to break the feedback loop to study and evaluate circuit behavior under open-loop conditions. One of the most important performance specifications determined from open-loop configurations is stability. When the feedback loop is properly broken then stability is easily determined from frequency response plots of the gain around the loop (known as loop-gain).
Quantitative measures of stability as well as approximate degrees of stability can be determined from performance parameters taken from the log magnitude and phase plots (Bode plots) of loop-gain. Gain margin is a term that refers to a factor by which the magnitude of loop-gain at a phase of minus one hundred eighty degrees (−180° must be changed in order to produce an unstable circuit. A gain margin greater than one (“1”) indicates a stable circuit while a gain margin less than one (“1”) indicates an unstable circuit.
Phase margin is a term that refers to the amount of phase shift at a loop-gain of unity (i.e., when the value of the loop-gain equals one (“1”)) that will cause the circuit to be unstable. A positive phase margin indicates a stable circuit while a negative phase margin indicates an unstable circuit.
The determinant (ΔY) of a nodal admittance matrix [Yn(s)] of a feedback circuit contains the poles of the closed-loop system. The determinant can be expressed in a form that reveals the “return ratio” (designated RR(s)) and the “return difference” (designated F(s)). Specifically,
                              Δ          γ                =                              det            ⁡                          [                                                Y                  n                                ⁡                                  (                  s                  )                                            ]                                =                                                    Δ                1                            +                              Δ                2                                      =                                                            Δ                  1                                ⁡                                  (                                      1                    +                                                                  Δ                        2                                                                    Δ                        1                                                                              )                                            =                                                                    Δ                    1                                    ⁡                                      [                                          1                      +                                              RR                        ⁡                                                  (                          ss                          )                                                                                      ]                                                  =                                                      Δ                    1                                    ⁢                                      F                    ⁡                                          (                      s                      )                                                                                                                              (        1        )            
The form shown in Equation (1) is possible if Δ1 and Δ2 are derived from response functions computed from the circuit with its feedback loop open. After these terms have been found, the polar plot of the return ratio may be investigated with the Nyquist criteria to determine closed-loop stability. Additional background material concerning this topic may be found in a textbook entitled Active Network Analysis by W. K. Chen published by World Scientific Publishing Company, Teaneck, N.J., 1991.
Several methods that are suitable for determining the return ratio from an open-loop circuit have been published. For specific examples, one may refer to H. W. Bode, Network Analysis and Feedback Amplifier Design, Van Nostrand, N.Y., 1945; F. H. Blecher, “Design principles for single loop transistor feedback amplifiers,” IRE Trans. Circ. Theory, Volume 4, No. 3, pp. 145-156, September 1957; R. D. Middlebrook, “Measurement of loop gain in feedback systems,” Int. J. Electron., Volume 38, pp. 485-512, 1975; S. Rosenstark, “Loop gain measurement in feedback amplifiers,” Int. J. Electron., Volume 57, pp. 415-421, 1984; P. J. Hurst, “Exact simulation of feedback circuit parameters,” IEEE Trans. Circ. Syst., Volume 38, pp. 1382-1389, November 1991; P. J. Hurst and S. H. Lewis, “Determination of stability using return ratios in balanced fully differential feedback circuits,” IEEE Trans. Circ. Syst. II, Volume 42, No. 12, pp. 805-817, December 1995; and H. T. Russell, Jr., “A loop-breaking method for the analysis and simulation of feedback amplifiers,” IEEE Trans. Circ. Syst., Volume 49, pp. 1045-1061, August 2002.
In the feedback amplifier textbook by H. W. Bode the return ratio is defined as a function of a pre-selected dependent source parameter “k”. With all independent sources removed from the circuit and with the chosen dependent source replaced by an independent source of the same type, the feedback loop is essentially broken allowing the feedback function f(s) to be computed. The return ratio is defined to be “k” times “f(s)”. One problem with this method occurs in the application of the method to circuits that use complex transistor models. The dependent sources in these models are often deeply embedded in such a way that they are not easily accessible for replacement and calculation of the feedback function. This is especially true for circuits that use the models in a software-based circuit simulator such as the SPICE circuit simulator. The SPICE circuit simulator is a well-known software program that simulates the operation of electrical and electronic circuits.
In the method proposed in the Middlebrook reference (cited above) and later refined in the Rosenstark reference (cited above) the feedback loop is physically broken at an arbitrary point and test signals are injected into the break. The signals (both current and voltage) are used to compute a pair of transfer functions that describe the short-circuit current gain and the open-circuit voltage gain around the loop. A single expression for the return ratio of the circuit is computed from a combination of these functions. A major problem with this method is the necessity of performing multiple measurements required to generate the two transfer functions.
Another problem that is likely to be encountered with the prior art methods is associated with their application in SPICE simulations. When the feedback loop is broken, the closed-loop direct current (dc) bias point is no longer maintained. Consequently, the small-signal alternating current (ac) models of transistors and other non-linear devices will not be accurate because these models must be determined at closed-loop direct current (dc) values. Fortunately, the problem can be solved by applying prior art replicate biasing methods. Typical prior art replicate biasing methods are described in the above cited references (1) by P. J. Hurst, (2) by P. J. Hurst and S. H. Lewis, and (3) by H. T. Russell, Jr. These prior art methods, however, require one or more of the following operations: (1) the replacement of a dependent source by an independent source of the same type, (2) multiple measurements and calculations performed to generate a single expression, and (3) the representation of components in the feedback loop as two-port networks. These prior art methods cannot be used if these operations cannot be performed (or are not performed).
Therefore, there is a need in the art for an improved system and method that is capable of breaking a feedback loop in a feedback circuit in which it is not necessary (1) to replace a dependent source with an independent source of the same type, (2) to perform multiple measurements and calculations for a single expression, and (3) to model components in the feedback loop as two-port networks.